Littrow resonators and their triply resonant nonlinear response

Among open resonators of mesoscopic size, we have shown recently that broad waveguides with Littrow diffraction at one of their boundaries could produce a set of sharp resonances in a silicon-on-insulator implementation at 1550 nm. We draw here two perspectives: firstly, we make use of a fundamental theory initially proposed in 1995 by Demkov and Ostrovsky, for Landau-Zener phenomena in the case of crossing manifolds, to explain the flatness of bands in such broad periodic waveguides. Secondly, we capitalize on the fact that longer waveguides behave as multiple series resonators to investigate the χ(3) nonlinear the triply resonant wavelength conversion process. The free-spectral-range of our waveguides are in the 15-30 nm range. Triply resonant conversion demands equal spacing, thus controlled dispersion, which could be achieved by playing with multiple resonators and the rest of the parametric space of these systems.